I have the solution, maybe, 80% chance this will work, but anyways... I figured something out, I hope this helps you. Today I was trying to fix game center by restarting it. So after about 9 tries, i opened game center and it worked! But I realized that I had no wifi or data so I turned it on and game center still didn't work. So I decided to try something. I forced restarted it (home button and power button for 10 sec), typed in my passcode, and turned on airplane mode, and when I opened my game center, to my relief, it worked! So I kept airplane mode on, went on settings and opened the game center option, I thought, it's probably going to freeze, but to my surprise it didn't! After about 10 seconds, it loaded, so I clicked logout. I stayed logged out and went back on game center. After about 7 seconds it asked me to log in so instead, I turned on my wifi again, logged in and boom, my game center worked fine! I was so happy! But after a while it will suddenly stop working so you will have to repeat the same process. Well I hope this works for all of you guys! If it doesn't I a very sorry for wasting your time. Well then, I'll see you guys! =)